Related Rates Calculus

Question

 Gravel is being dumped from a conveyor belt at a rate of 20 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 19 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V = 1/3 π r²h. When the pile is 19 feet high, its height is increasing at ___________feet per minute.

Answer:

Answer 1

Gravel is being dumped from a conveyor belt at a rate of 20 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 19 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V = 1/3 π r²h. When the pile is 19 feet high, its height is increasing at ___________feet per minute.

Given:

V' = 20, 2r = h → r = h/2, V = π/3r²h → V = π/3(h/2)²h = (π/12)h³

Find h' when h = 19

Power Rule: d(xⁿ)/dx = nx^n-1

V'(h) = (π/12)(3h²h') = 20

h' = 80/(h²π) = 80/(19²π) ≅0.07 ft/min